Unit 3.2: Implicit Differentiation

What is Implicit Differentiation?

Sometimes, equations are given implicitly, meaning that y is not isolated such as:


How do we differentiate this?

The general steps to implicitly differentiate is:

  • Differentiate both sides with respect to the variable

    • Note: certain problems like related rates, requires the equation to be differentiated with respect to time or another variable

    • Use Chain rule when involving another variable other than the variable of differentiation

  • Rewrite the equation to isolate dy/dx 

  • Substitute y in terms of x if asked 

Let’s use the example on the right and bottom to practice.

Basic Walk through:

1) Differentiate both sides with respect to x:

2) Solve for your new derivative:

3) Because we are asked to solve in terms of x:

  • Solve for y from the original equation

4) Substitute y in for the equation:

Practice Problems

 

Differentiate the following equation in terms of x and y:

Basic Walk through:

1) Differentiate both sides with respect to x:

2) Solve for your new derivative:

3) Because we are asked to solve in terms of x and y, we dont need to substitute

Differentiate the following equation in terms of x: